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9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 Write and test a recursive function that returns the sum of the squares of the first n positive integers. Write and test a recursive function that returns the sum of the first n powers of a base b. Write and test a recursive function that returns the sum of the first n elements of an array. Write and test a recursive function that returns the maximum among the first n elements of an array. Write and test a recursive function that returns the maximum among the first n elements of an array, using at most lg n recursive calls. Write and test a recursive function that returns the power x n. Write and test a recursive function that returns the power xn, using at most 2 lg n recursive calls. Write and test a recursive function that returns the integer binary logarithm of an integer n (i.e., the number of times n can be divided by 2). Write and test a recursive boolean function that determines whether a string is a palindrome. (A palindrome is a string of characters that is the same as the string obtained from it by reversing its letters.)
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9.10 Write and test a recursive function that returns a string that contains the binary representation of a positive integer. 9.11 Write and test a recursive function that returns a string that contains the hexadecimal representation of a positive integer. 9.12 Write and test a recursive function that prints all the permutations of the first n characters of a string. For example, the call print("ABC",3) would print
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ABC ACB BAC BCA CBA CAB
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9.13 Implement the Fibonacci function iteratively (without using an array).
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RECURSION
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9.14 Implement the recursive Ackermann function: A(0, n) = n + 1 A(m, 0) = A(m 1, 1), if m > 0 A(m, n) = A(m 1, A(m, n 1)), if m > 0 and n > 0 9.15 Prove Pascal s recurrence relation (page 170). 9.16 Trace the recursive implementation of the Euclidean Algorithm (Example 9.13 on page 171) on the call gcd(385, 231). 9.17 Implement the Euclidean Algorithm (page 171) iteratively. 9.18 Implement the recursive Euclidean Algorithm using the integer remainder operator % instead of repeated subtraction. 9.19 Implement the Euclidean Algorithm iteratively using the integer remainder operator % instead of repeated subtraction. 9.20 Use mathematical induction to prove that the recursive implementation of the Fibonacci function (Example 9.6 on page 167) is correct. 9.21 Use mathematical induction to prove that the recursive function in Problem 9.4 is correct. 9.22 Use mathematical induction to prove that the recursive function in Problem 9.5 is correct. 9.23 Use mathematical induction to prove that the recursive function in Problem 9.8 is correct. 9.24 Use mathematical induction to prove that the recursive function in Problem 9.12 is correct. 9.25 The computable domain of a function is the set of inputs for which the function can produce correct results. Determine empirically the computable domain of the factorial function implemented in Example 9.2 on page 165. 9.26 Determine empirically the computable domain of the sum(b,n) function implemented in Problem 9.2 on page 177, using b = 2. 9.27 Determine empirically the computable domain of the Fibonacci function implemented in Example 9.3 on page 166. 9.28 Determine empirically the computable domain of the recursive binomial coefficient function (Example 9.11 on page 170). 9.29 The Towers of Hanoi program performs 7 disk moves for 3 disks. How many disk moves are performed for: a. 5 disks b. 6 disks c. n disks 9.30 Prove the formula that you derived in previous problem. 9.31 Determine empirically the computable domain of the Ackermann function (Problem 9.14). 9.32 Show the recursive call tree for the call hanoi(4,'A','B','C') in Example 9.15 on page 174. 9.33 Modify the program in Example 9.16 on page 175 so that the results are more accurate by narrowing the bases so that recursion continues until | x | < 0.00005. 9.34 Modify the program in Example 9.16 on page 175 so that the results are obtained in fewer iterations by using the more accurate Taylor approximations sin x x x 3/6 + x5/120 cos x 1 x 2/2 + x4/24
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